Question

Question:Show that additive group $\mathrm{\mathbb{Z}}and\mathrm{\mathbb{Q}}$are not isomorphic.

Short answer

Answer

Additive group $\mathrm{\mathbb{Z}}and\mathrm{\mathbb{Q}}$are not isomorphic

Step-by-step solution

Definition of Isomorphism 

$$\begin{gathered} \text{LetGandHbegroupswithagroupoperationdenotedby*,} \\ \text{GisisomorphictoagroupH(insymbolsG≅H)} \\ \text{ifthereisafunctionf:→Hsuchthat,} \\ 1.If(a*b)=f\left(a\right)*f\left(b\right) \\ 2.fisinjective. \\ 3.fissurjective. \end{gathered}$$

Proving that the additive group ℤ and ℚ are not isomorphic

Suppose that additive groups $\mathrm{\mathbb{Z}}and\mathrm{\mathbb{Q}}$are isomorphic.

Therefore,$\varphi :\mathrm{\mathbb{Z}}\to \mathrm{\mathbb{Q}}$

So, there should be some $r\in \mathrm{\mathbb{Q}}$, such that

$$\begin{gathered} \varphi \left(r\right)=1_z \\ \varphi \left(r\right)=\varphi (\raisebox{1ex}{$r$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+\raisebox{1ex}{$r$}\!\left/ \!\raisebox{-1ex}{$2$}\right.) \\ =2\varphi \left(\raisebox{1ex}{$r$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right) \\ so,\varphi (\raisebox{1ex}{$r$}\!\left/ \!\raisebox{-1ex}{$2$}\right.)=\frac{1}{2} \end{gathered}$$

Since is an integer, so $\varphi (\raisebox{1ex}{$r$}\!\left/ \!\raisebox{-1ex}{$2$}\right.)=\frac{1}{2}$cannot be in $\mathrm{\mathbb{Z}}$.

Hence, additive groups $\mathrm{\mathbb{Z}}and\mathrm{\mathbb{Q}}$are not isomorphic.